A new approach for multistep numerical methods in several frequencies for perturbed oscillators
Introduction
Forced oscillators are presented in many Physics and Engineering models. Harmonic oscillators also form part of Celestial Mechanics models, such as the classic two-body problem, and the satellite problem, as the K-S and B-F transformations reduce the Kepler problem to oscillators. Efficient numerical algorithms are needed, therefore, to provide very precise approximations.
In [1], the so called Ferrándiz φ- functions are defined, according to their properties. Based on these, a series method is constructed in order to numerically integrate perturbed, stable and convergent oscillators. Said method is a generalisation of the Scheifele series methods [2], [3] which, with two frequencies integrates the non-perturbed problem exactly. Recently a new method, TFSTS, based on the Scheifele methods, which verify this property has been published in [4]. The series method is extremely precise. However, it has the disadvantage that it needs to be adapted to each specific problem. In order to resolve this difficulty, in [5], the transformation of the φ-function series method in the MDFpPC multi-step scheme is explained. Calculation of the coefficients of the multi-step scheme are made through a recurrent procedure based on the existing relation between the divided differences and the elemental and complete symmetrical functions.
The recurrent calculation of the coefficients permits the MDFpPC scheme to be considered as a VSVO type scheme.
The algorithm proposed in [5], despite its good behaviour, presents the difficulty of imprecise integration of the homogenous problem. In addition, it requires the definition of two multi-step schemes both in the explicit and implicit case, according to the parity of the number of steps, a fact which makes it difficult to implement in a computer.
This article introduces a new proposal for the MDFpE, MDFpI and MDFpPC schemes in order to obtain a single formulation of the algorithms independently of the parity of the number of steps.
The new algorithm, while retaining the excellent properties of the aforementioned dual frequency methods, improves its precision, which is demonstrated by overcoming the problems proposed in [5] using both multi-step methods.
Section snippets
φ-functions series method
In this point a brief description is provided of the construction of the φ-functions and the corresponding numerical series method for integration of perturbed oscillators [1], [5].
With x(t) being the solution of the perturbed oscillator of the equationcorresponding to a forced oscillation with frequency α2 and perturbation function f = f(x(t), x′(t), t), from which it is supposed that it is analytic and with continuous partial derivatives with
A new approach of explicit method MDFpE for perturbed oscillators
As described in [5] we will substitute the derivatives by expressions in term of divided differences and next to some coefficients dij, elements of a matrix , of those we do not know a recurrence relation. Once the matrix is know, we will set up a recurrent calculus, through matrix Sp,n for the explicit method. The study of symmetric polynomials [6] and its relation with the divided differences, will allow us the computation of the matrix Sp,n.
To make a variable step explicit multistep
A new approach of implicit method MDFpI for perturbed oscillators
Similarly for the implicit case, the matrix of that we extract the coefficients dij we will denote as . The matrix, Bp as described in [5], [8], [10], is:
Designating by , it can write:
To obtain a new method MDFpI based on algorithm which described in [5], substituting (52) in (20) it obtains by
A new approach of predictor corrector method MDFp PC for perturbed oscillators
The predictor–corrector method modified, with variable step size, of p steps for perturbed oscillators is defined like the one which have as predictor the method MDFpE modified and as corrector MDFpI modified, with the previous definitions.
Numerical experiments
In this section, we present an application of the multistep method modified, to the resolution of stiff and highly oscillatory problems. The good behaviour of the method is presented by comparison with other known codes, implemented in the program packaged NUMERIC dsolve MAPLE, such as:
LSODE methods, causes a numerical solution to be found using the Livermore Stiff ODE solver.
GEAR causes a numerical solution to be found by way of a Burlirsch-Stoer rational extrapolation method.
MGEAR [msteppart]
Conclusions
A new VSVO type multi-step method has been constructed based on the φ-functions, which generalise the Scheifele G-functions.
The new algorithm integrates the oscillator without discretisation error, that is, if the perturbation terms disappear in an arbitrary instant of the independent variable t, the new method integrates the non-perturbed problem exactly. In the perturbed oscillators the perturbation parameterε, appears as a factor in the local truncation error.
Explicit, implicit and predictor
Acknowledgements
This work has been supported by GRE09-13 project of the University of Alicante, and the project of the Generalitatr Valenciana GV/2011/032.
References (13)
- et al.
Multistep numerical methods for the integration of oscillatory problems in several frequencies
Adv Eng Software
(2009) - et al.
Numeric multistep variable methods for perturbed linear system integration
Appl Math Comput
(2007) A new numerical method for the integration of highly oscillatory second-order ordinary differential equations
Appl Numer Math
(1993)- et al.
Accurate numerical integration of perturbed oscillatory systems in two frequencies
Trans Math Software TOMS
(2009) - et al.
Linear and regular celestial mechanics
(1971) On numerical integration of perturbed linear oscillating systems
ZAMP
(1971)
Cited by (1)
Multistep variable methods for exact integration of perturbed stiff linear systems
2016, Journal of Numerical Mathematics